To find the quotient of the given rational expressions, we need to follow these steps:
1. Rewrite the division as multiplication by the reciprocal:
The given problem is:
[tex]\[
\frac{3x - 6}{x^3} \div \frac{x - 2}{2x - 1}
\][/tex]
Division of fractions is equivalent to multiplying by the reciprocal of the divisor. Therefore, we rewrite it as:
[tex]\[
\frac{3x - 6}{x^3} \times \frac{2x - 1}{x - 2}
\][/tex]
2. Multiply the numerators and the denominators:
- The numerator of the first fraction is [tex]\(3x - 6\)[/tex].
- The denominator of the first fraction is [tex]\(x^3\)[/tex].
- The numerator of the second fraction is [tex]\(2x - 1\)[/tex].
- The denominator of the second fraction is [tex]\(x - 2\)[/tex].
So the new fraction after multiplication becomes:
[tex]\[
\frac{(3x - 6)(2x - 1)}{(x^3)(x - 2)}
\][/tex]
3. Simplify the expression:
- First, factor out common terms if possible. Notice that [tex]\(3x - 6\)[/tex] can be factored as [tex]\(3(x - 2)\)[/tex].
Substituting this factorization in, the expression becomes:
[tex]\[
\frac{(3(x - 2))(2x - 1)}{(x^3)(x - 2)}
\][/tex]
- We see that [tex]\((x - 2)\)[/tex] appears in both the numerator and the denominator, so we can cancel [tex]\((x - 2)\)[/tex] out:
[tex]\[
\frac{3(2x - 1)}{x^3}
\][/tex]
Simplifying this, we get:
[tex]\[
\frac{6x - 3}{x^3}
\][/tex]
4. Compare the options:
After simplifying, we need to match our reduced form with one of the given options:
A. [tex]\(\frac{4x - 8}{x^3 + 2x - 1}\)[/tex]
B. [tex]\(\frac{5x - 7}{x^3 + x - 2}\)[/tex]
C. [tex]\(\frac{3x^2 - 12x + 12}{2x^4 - x^3}\)[/tex]
D. [tex]\(\frac{6x - 3}{x^3}\)[/tex]
E. [tex]\(\frac{6x^2 - 15x + 6}{x^4 - 2x^3}\)[/tex]
Only one of these options matches our simplified expression:
[tex]\[
\boxed{D}
\][/tex]
